Singular perturbation with Hopf points in the fast dynamics

A singular perturbation problem in ordinary differential equations is investigated without assuming hyperbolicity of the associated slow manifold. More precisely, the slow manifold consists of a branch of stationary points which lose their hyperbolicity at a stationary point with eigenvalues ii. Thus, it is impossible to reduce the dynamics near the Hopf point to the slow manifold. Tills situation is examined within a generic one parameter unfolding. It leads to two bifurcating curves of Hopf points and associated to these are two manifolds of periodic orbits and possibly another manifold of invariant toll, all three of which intersect in the central Hopf point. The proof employs a suitable Ansatz resulting ill a Hopf bifurcation theorem which determines precisely the bifurcation structure near a certain Hopf point with an additional zero eigenvalue.